The Spanish boss guided the club to FA Cup glory against Manchester City, but failed to retain their Premier League status after suffering defeat against Arsenal just three days later.
Martinez has so far been coy on his future at the club, and, as exclusively revealed by Goal, has been identified as Everton's No. 1 target to replace new Manchester United boss David Moyes.
|11/2||Roberto Martinez is 11/2 with Coral to be the next Everton manager
"Roberto wants 24 hours to think about what is going on," Whelan told BBC Sport. "Tomorrow is D-Day.
"I had a meeting with Roberto at noon today. It was a very constructive meeting where we discussed all kinds of things - winning the Cup, our relegation, our prospects for next season.
"Roberto wants 24 hours to think about what is going on. He wants to speak to his wife, think about it, which I fully understand.
"We are going to meet tomorrow and Roberto is going to make his mind up whether he wants to continue his contract with Wigan or whether he wants to take up a new position elsewhere.
"There is a massive amount of loyalty between Roberto and myself. If he wants to stay, that is fantastic and I will be over the moon. But if decides he wants to move on I will full respect that and I will give him every bit of support I can."